Abstract

We show that the Fréchet derivative of a
matrix function $f$ at $A$ in the direction $E$,
where $A$ and $E$ are real matrices,
can be approximated by
$\Im f(A+ihE)/h$
for some suitably small $h$.
This approximation,
requiring a single function evaluation at a complex argument,
generalizes the complex step approximation known in the
scalar case.
The approximation is proved to be of second order in $h$
for analytic functions $f$ and also for
the matrix sign function.
It is shown that it does not suffer the inherent cancellation
that limits the accuracy of finite difference
approximations in floating point arithmetic.
However,
cancellation does nevertheless vitiate the approximation
when the underlying method for evaluating $f$ employs
complex arithmetic.
The ease of implementation of the approximation,
and its superiority over finite differences,
make it attractive when specialized methods for
evaluating the Fréchet derivative are not available,
and in particular for condition number estimation
when used in conjunction with a block 1-norm estimation algorithm.